Topics in Topology ("Topologie III"), Sommersemester 2025

lecture notes
The actual notes for this course begin with Chapter 56, on page 493.
What appears before Chapter 56 is a complete set of notes that I wrote for Topology I-II in previous semesters. This is convenient to have there so that I can occasionally refer to it in the current course, but that does not mean that you must already know most of what is contained in the earlier parts of the notes. (See Prerequisites below for more on that.)

Problem Sets

Weekly problem sets will be posted here, starting in the first week of the semester.

For general information about the course, scroll down a bit...


image by Christian Lawson-Perfect from cp's mathem-o-blog


General information

Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage)

Moodle:
https://moodle.hu-berlin.de/course/view.php?id=133147
The enrollment key is: obstruction
Important: You must join the moodle for the course in order to receive occasional time-sensitive announcements, e.g. if a lecture has been cancelled or rescheduled. HU students can access moodle using their HU username and password. Non-HU users can access it by following this link and then clicking on "Create new account". You will need to enter the enrollment key printed above.

Time and place:
Lectures on Wednesdays and Thursdays 11:15-12:45 in room 1.012 (RUD25)
Problem Class (Übung) Wednesdays 13:15-14:45 in room 1.012 (RUD25)
Note: We may adjust the Wednesday schedule a bit in order to allow for a more comfortable lunch between lecture and problem class. This will be discussed and decided in the first lecture.

Language:
The course will be taught in English.

Prerequisites:
The main prerequisites are a solid foundation in point-set topology, the fundamental group and covering spaces, singular and cellular (co-)homology (including computations based on the axioms and some homological algebra, e.g. the universal coefficient theorems), and some willingness to put up with the language of categories, functors and universal properties. If you have taken my Topologie II course before, then you definitely have the essential prerequisites, but you probably also have them if you learned about homology and cohomology elsewhere. Some knowledge of smooth manifolds will occasionally be useful, but if you do not have this, you will just need to be willing to accept a small set of facts about tangent spaces, tubular neighborhoods and transversality as black boxes.

Contents:
This is a course on intermediate-level algebraic topology, and is conceived in part as a sequel to last semester's Topology 2 course. We aim to prove some useful results from elementary homotopy theory (some of which were mentioned briefly last semester but were not proved), and introduce the essentials of obstruction theory, classifying spaces, characteristic classes, and bordism theory. These are all topics that have wide-ranging applications to other areas of mathematics, especially to problems in differential geometry and topology, such as the existence and classification of exotic smooth structures on manifolds. We will not attempt any deep exploration of modern homotopy theory, as that would far exceed my expertise.

Here is a more detailed plan of topics, though I cannot promise that all of them will be covered.

Literature:
The top of this page contains a link to detailed lecture notes for this course which will be updated routinely as the course progresses. The bulk of what we plan to cover is in any case contained in the union of the following three books:

Each of these books has its own set of advantages and disadvantages, and none contains absolutely everything that I'm hoping to cover; each also contains plenty of interesting stuff that we will not have time for. Here are some other sources that are good to know about: Here are some references that we are less likely to make much direct use of, but it would be wise to remember that they are also out there:

Homework:
Weekly problem sets will be posted near the top of this page, and some subset of those problems will be discussed in the problem class (Übung) each week. Homework will not be collected or graded. The problem class may sometimes also be used to fill in gaps on details that did not fit into the regular lectures.

Grades:
Since this is an advanced course, I have a fairly relaxed attitude about grades. If you come to the course with adequate prerequisites and stay with it for the whole semester, you can come to my office at the end for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular coherent topic from the course to focus on, typically the contents of four to six lectures, and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.

Chris Wendl's homepage